Landau limit

This is the Chandrasekhar-Landau limiting mass for white dwarfs, whose actual value (derivable from the theory of polytropic stars see Appendix 4) is... 

The above demonstrates therefore that the Gaussian approximation at Eq. (30) correctly interpolates between the Coulomb and Landau limits. This, needless to say, does not mean that Eq. (30) is exact throughout the entire span of magnetic field intensities. However, it can be taken as encouraging for the utility of the semiclassical expansion, after higher-order corrections have been incorporated. 

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

The basic Landau-Ginzburg model is valid only for relatively weak surfactants and in a limited region of the phase space. In order to find a more general mesoscopic description, valid also for strong surfactants and in a more extended region of the phase space, we derive in this section a mesoscopic Landau-Ginzburg model from the lattice CHS model [16]. 

Landau then identifies turbulence as the limit state consisting of an infinite number of incommensurate frequencies. 

It gives 1 and dir/Lz in the limit of large and small values of the Landau-Zener parameter, respectively. 

The Landau-Zener formula in the limit J2/ v - oo gives PLZ = 1 whereas Eq. (183) for the case of fast fluctuations in this limit gives P — At small values of J2/ u, both formulas give the same result, P = 2itJ2/ v. ... 

For some unknown reason Chandrasekhar (1931) took p equal to the molecular weight, 2.5, for a fully ionized material , corresponding to K = 3.619-1014 and obtained the limiting mass from (1) equal to 1.822 1033g= 0.91 M . Landau (1932) noticed, that for most stable nuclei He4, C12, O16 etc. the value of //. which is the number of nucleons to one electron, is equal to 2, and he obtained the realistic value of the limiting mass Mum = 1.4 M . 

DR. MEYER Dr. Schatz has led me to believe in certain things, and I want to ask him a question. It occurs to me that the Landau-Zener equation cannot work except in the nonadiabatic limit. Do you have any comments on that ... 

DR. MARSHALL NEWTON (Brookhaven National Laboratory) The Landau-Zener theory in the weak-coupling regime may be derived by perturbation theory, as Dr. Sutin mentioned earlier. The complete derivation is much more general than the perturbation theory, and allows one to include an arbitrary degree of coupling. Thus, one can go continuously from nonadiabatic to the adiabatic limits. 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

It is clear from eq. (17) that A describes how the Ginzburg-Landau functional F[8Mq] varies with q. Here, 8Mq are the Fourier components of the difference between local and macroscopic magnetization M(T, H). In the long-wave limit, in which eq. (17) is valid. 

The classical Liouvillian operator Zc, which is the classical limit of the Landau-von Neumann superoperator in Wigner representation, can also be analyzed in terms of a spectral decomposition, such as to obtain its eigenvalues or resonances. Recent works have been devoted to this problem that show that the classical Liouvillian resonances can be obtained as the zeros of another kind of zeta function, which is of classical type. The resolvent of the classical Liouvillian can then be obtained as [60, 61]... 

In the classical limit h - 0, the spectrum of the Landau-von Neumann superoperator tends to the spectrum of the classical Liouvillian operator. If the classical system is mixing, the classical Liouvillian spectrum is always continuous so that we may envisage an analytic continuation to define a discrete spectrum of classical resonances. It has been shown that such classical resonances are given by the zeros of the classical zeta function (2.44) and are called the Pollicott-Ruelle resonances sn(E) [63], These classical Liouvillian resonances characterize exponential decay and relaxation processes in the statistical description of classical systems. The leading Pollicott-Ruelle resonance defines the so-called escape rate of the system,... 

Landau also gives solutions of Eqs. (99) to (103) for the limiting cases... 

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